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Chapter 19: Problem 1888

If \(\cos ^{-1} x-\sin ^{-1} x=(\pi / 4)\) then \(x=\) (a) \([\sqrt{\\{} 2-\sqrt{2}\\} / 2]\) (b) \([\sqrt{\\{2}+\sqrt{2}\\} / 2]\) (c) \(\sqrt{2}-1\) (d) \(\sqrt{2}+1\)

Short Answer

Expert verified
The short answer is: \(x = [\sqrt{2 - \sqrt{2}} / 2]\) (Option a).

Step by step solution

01

Since we know that \(\sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2}\), we can substitute \(\sin^{-1}(x)\) with the expression in terms of \(\cos^{-1}(x)\). So, \(\sin^{-1}(x) = \frac{\pi}{2} - \cos^{-1}(x)\) Now, we can replace \(\sin^{-1}(x)\) in the given equation: \(\cos^{-1}(x) - (\frac{\pi}{2} - \cos^{-1}(x)) = \frac{\pi}{4}\) #Step 2: Solve for cos inverse x#

Simplify the equation to find the value of \(\cos^{-1}(x)\): \(2\cos^{-1}(x) - \frac{\pi}{2}= \frac{\pi}{4}\) Now, add \(\frac{\pi}{2}\) to both sides and divide by 2: \(\cos^{-1}(x) = \frac{\pi}{4} + \frac{\pi}{8} = \frac{3\pi}{8}\) #Step 3: Find x using cosine inverse#
02

Now, we can find the value of \(x\) by taking the cosine of both sides of the equation: \(x = \cos(\frac{3\pi}{8})\) Now we can use the half-angle identity to find the value of the cosine function: \(\cos(\frac{3\pi}{8}) = \sqrt{\frac{1 + \cos(\frac{3\pi}{4})}{2}}\) Since \(\cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2}\), we can replace it in the equation: \(x = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}\) #Step 4: Simplify the expression#

Now, we just need to simplify the expression for \(x\): \(x = \sqrt{\frac{2 - \sqrt{2}}{4}}\) Upon comparing the expression to options (a), (b), (c), and (d), we find the answer is: \(x = [\sqrt{2 - \sqrt{2}} / 2]\) (Option a).

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